Criteria for Local Tabularity of Modal Logic Products

核心概念

The core message of this article is to provide semantic and axiomatic conditions that characterize when the product of two locally tabular modal logics is itself locally tabular.

要約

The article investigates the local tabularity of products of modal logics. It makes the following key observations:

1. Local tabularity of the factors L1 and L2 is necessary but not sufficient for local tabularity of the product L1 × L2. The simplest counterexample is the logic S5 × S5, which is not locally tabular despite both factors being locally tabular.

2. The authors provide extra semantic and axiomatic conditions that give criteria for local tabularity of the product of two locally tabular logics:

   • Bounded cluster property of one of the factors
   • A condition called "product reducible path property"
   • Finiteness of the one-variable fragment of the product

3. They apply these criteria to identify new families of locally tabular products, including:

   • Products of logics above S4 containing formulas of finite height
   • Products of locally tabular logics with S5
   • Products of locally tabular logics with the logic Tack (the logic of the ordered sum of a countable cluster and a singleton)

4. The article also gives a semantic argument for the known result that all proper extensions of S5 × S5 are locally tabular.

統計

None.

引用

None.

抽出されたキーインサイト

Locally tabular products of modal logics

by Ilya B. Shap... 場所 arxiv.org 04-03-2024

https://arxiv.org/pdf/2404.01670.pdf

深掘り質問

Can the criteria developed in this article be extended to characterize local tabularity of products of more than two modal logics

The criteria developed in the article can potentially be extended to characterize the local tabularity of products involving more than two modal logics. By considering the interactions between the individual logics in the product and their respective properties such as bounded cluster property, reducible path property, and 1-finiteness, it may be possible to create a framework that can assess the local tabularity of products involving multiple modal logics. However, the complexity of such extensions would increase with the number of logics involved, requiring a more intricate analysis of their interactions and properties.

Are there other semantic or syntactic properties of modal logics, beyond the ones considered here, that can guarantee local tabularity of their products

In addition to the semantic and syntactic properties discussed in the article, there are other characteristics of modal logics that could potentially guarantee the local tabularity of their products. For example, properties related to the frame conditions, such as the existence of certain types of relations or structures within the frames, could play a role in determining the local tabularity of the products. Additionally, properties related to the expressive power of the modal logic, such as the presence of specific axioms or inference rules, could also impact the local tabularity of the products.

What are the connections between local tabularity of modal logic products and other important properties, such as the finite model property or decidability

The local tabularity of modal logic products is closely connected to other important properties such as the finite model property and decidability. The finite model property ensures that a logic has models of finite size for every satisfiable formula, which can impact the local tabularity of its products by constraining the complexity of the frames involved. Decidability, on the other hand, relates to the ability to algorithmically determine the truth of formulas in a logic, which can influence the analysis and verification of local tabularity in modal logic products. Understanding these connections can provide insights into the broader implications of local tabularity in modal logic.